Proposed Problem

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Complete Problem 288: Tangent circles, Harmonic Mean, Radius, Diameter

Collection of Geometry Problems

Level: High School, SAT Prep, College geometry

## Saturday, May 9, 2009

### Problem 288: Tangent circles, Harmonic Mean, Radius, Diameter

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This is an excellent problem that had me flummoxed for days together. Finally a friend who is simply brilliant at math, particularly plane geometry, made the suggestion that Apollonius Theorem applied to triangle AEB readily gives the result. So thanks to Oscar Rojas of Costa Rica we've: AE=R+x,BE=r+x,CE=(R+r)/2 - x while AC=CB=(R+r)/2. So we can say:

ReplyDelete(R+x)^2+(r+x)^2=2[(R+r)/2 - x]^2 + (R+r)^2/2

From where we get x = R*r/2(R+r)=2R*r/4(R+r)=1/4*H(R,r)

Ajit: On behalf of Oscar Rojas, Costa Rica

Median length, Apollonius' Theorem at:

ReplyDeletehttp://www.gogeometry.com/geometry/median_length_apollonius_theorem.htm